How to Identify Trigonometric Identities Using De Moivre’s Theorem for ( cos 4a 8sin^4a - 8sin^2a 1 )
Introduction to De Moivre’s Theorem
De Moivre’s theorem is a fundamental concept in complex analysis that connects trigonometric functions and complex numbers through the identity:
(1) Ze^{itheta} costheta isintheta)
By raising both sides to the power of 4, we can use this theorem to derive various trigonometric identities, including the one involving ( cos 4a ).
Applying De Moivre’s Theorem to ( e^{i4a} )
Starting with the identity:
(2) e^{i4a} (cos a isin a)^4
We can expand the expression on the right-hand side using the binomial theorem:
(3) (cos a isin a)^4 cos^4a 4icos^3asin a - 6cos^2asin^2a - 4icos asin^3a sin^4a
Separating the real and imaginary parts, we get:
Real part: ( cos^4a - 6cos^2asin^2a sin^4a )
Imaginary part: ( 4cos^3asin a - 4cos asin^3a )
Extracting the Real Part for ( cos 4a )
Focusing on the real part, we have:
(4) cos 4a cos^4a - 6cos^2asin^2a sin^4a
Further simplifying the expression:
(5) cos 4a (cos^2a sin^2a)^2 - 8cos^2asin^2a
Since ( cos^2a sin^2a 1 ), we can substitute this value in:
(6) cos 4a 1 - 8cos^2asin^2a
Using the double-angle identity ( cos 2a 2cos^2a - 1 ), we get:
(7) cos^2a frac{1 cos 2a}{2}
Substituting this into the equation:
(8) cos 4a 1 - 8left(frac{1 cos 2a}{2}right)left(frac{1 - cos 2a}{2}right)
(9) cos 4a 1 - 8left(frac{(1 cos 2a)(1 - cos 2a)}{4}right)
(10) cos 4a 1 - 2(1 - cos^2 2a)
(11) cos 4a 1 - 2(1 - sin^2 2a)
(12) cos 4a 1 - 2 2sin^2 2a
(13) cos 4a 8sin^4a - 8sin^2a 1
This completes the derivation and simplification of the trigonometric identity using De Moivre’s theorem.
Conclusion
De Moivre’s theorem provides a powerful tool for simplifying and deriving trigonometric identities. By understanding and applying this theorem, one can efficiently manipulate and identify complex trigonometric expressions. The identity ( cos 4a 8sin^4a - 8sin^2a 1 ) is a prime example of how De Moivre’s theorem can be used to derive and simplify such identities.